It really depends on what kind of experience you're looking for with "Death Note." If you prefer a more in-depth look at the characters and plot intricacies, I’d recommend starting with the manga. The original creator, Tsugumi Ohba, does an incredible job of exploring Light Yagami's psychological transformation and the moral questions surrounding justice and power. The pacing in the manga allows for more character development and subtle themes that sometimes get rushed in the anime adaptation.
On the other hand, the anime is visually stunning and has a captivating soundtrack that heightens the tension and drama. The voice acting is excellent as well, particularly Mamoru Miyano as Light and Noriko Hidaka as Misa. The anime condenses a lot of the material, making it a more thrilling, fast-paced experience, but some nuances from the manga get lost.
If you're a completer and really want the full picture, read the manga first; you won’t miss any plot points or character arcs. However, if you're looking for something that's immediately engaging and visually stimulating, start with the anime. Afterward, you can always circle back to the manga for a deeper dive.
Both mediums are fantastic, and you can't really go wrong with either!